Hook-partition formulae for Kadar–Yu Gram determinant polynomials

Let $l$, $l'$, and $l”$ be integer parameters, and let $C^{(\lambda)}(\alpha)$ and $P_n^{(\lambda)}(\alpha)$ denote the factors in the Kadar–Yu Gram determinant formula for a partition $\lambda$. For the hook partitions $(l'+1,1)$ and the transpose $(l+1,1)^T$, the displayed expressions give their $C$-factors and Chebyshev expansions.

Hook-partition formula conjecture. Formulae for $C^{(l'+1,1)}(\alpha)$, $C^{(l+1,1)^T}(\alpha)$, $P^{(l”+1,1)}_{l”+4+k}(\alpha)$, and $P^{(l+1,1)^T}_{l+4+k}(\alpha)$ hold for all $l\geq 1$, $l'\geq 4$, and $l”\geq 2$.

The formulas were obtained from direct computations in bounded ranges and, if valid uniformly, would extend the determinant description to all the indicated hook-partition parameters.